Crack Paths 2009

where ∂xjδ(r,θ) = ∂xjδ(x) ∣ ∣

(θ) and ψ is the monotone increasing and positive

x ∈ Υ h

function

√ h − r √ r + h a r c t(a n√ √ r

thus 0 ≤ψ(r)≤ π2h.

ψ(r) =

h − r)

(see condition (6)) can be calculated ex

The function C(θ) = −4√2π (̂A(xtip)−1 ) 11

actly and depends on the function δ as well as on the crack shoot Υh(θ). Finally,

we get

( ˜Φj,1i(A;θ) + ˜Φj,1i(B;θ)π2δ(xtip) ) h

∆ U≈ C(θ) i2,j=1∑

Kh3−iK0j (

(9)

−˜Φj,1i(B;θ)

h ∫

dr

θ)

ψ(r)(cos(θ)∂x1δ(r,θ)+ sin(θ)∂x2δ(r,

)

) + O(h3/2).

0

Because the function δ is smooth and bounded, the integral in (9) can be estimated

and is of order h. But this would be a loss of information. In principle, this integral

is the change of δ along the shoot Υh(θ). All terms can be calculated for any smooth

function δ and any angle θ. This formula is a first step to detect the influence of

local inhomogeneities on the crack path.

E X A M P LAENSCD O N C L U S I O N S

Finally, we show first numerical results and consider a symmetric compact tension

(CTS-)specimen subjected to a Mode-II-loading (see Figure 1). The length units

are selected to w = 90mm,the thickness of the specimen is 1 0 m mand we apply

a force F = 10000N. W echoose a local gradation only in one space direction (see

Fig. 2):

(x1−20)5π )

20.

0.5sin (

δ(x) := {

A ( x ) = ( 1δ(+x))A,

0

otherwise

Because this function is not smooth, we flatten out A(x) at the points x1 = 20 and

x1 = 25. This is only technical and we go not into details. The elastic moduli are

a11 = a22 = λ + 2µ, a21 = λ, a31 = a32 = 0, a33 = µ with λ = 56023N/mm2,µ =

26364N/mm2,corresponding to aluminium alloy 7075 − T651. Our motivation of

this simple example is just: "Whatcan happen, if the material is locally functionally

graded?"

Numerical computations are done with deal.II (dealii.org) and the meshgenerator

Cubit 11.0 (cubit.sandia.gov). To calculate SIFs, we use weight functions and solve a

pure N e u m a npnroblem without clamping the specimen, see [2, 7] for more details.

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